NTA UGC NET JUNE-2023 Paper-2

Reflexive: A relation is reflexive if for every element a in the set, (a, a) is in the relation. In this case, if we take a = b = c = d, we have a * d(b + c) = b * c(a + d), which simplifies to a * d(a) = a * a(a + d), which is true. So, the relation is reflexive.

Symmetric: A relation is symmetric if for every (a, b) in the relation, (b, a) is also in the relation. In this case, if (a, b) is in the relation, then a * d(b + c) = b * c(a + d). If we swap (a, b) to (b, a), we have b * c(a + d) = a * d(b + c), which is equivalent to the original equation. So, the relation is symmetric.

Transitive: A relation is transitive if whenever (a, b) and (b, c) are in the relation, (a, c) must also be in the relation. In this case, if (a, b) and (b, c) are in the relation, it means that both of these equations hold: a * d(b + c) = b * c(a + d) and b * d(c + a) = c * a(b + d). Now, if we multiply these two equations, we get a * d(b + c) * b * d(c + a) = b * c(a + d) * c * a(b + d), and after simplification, we get a * c(b + c) = b * c(a + d), which is the same as the first equation. So, the relation is transitive.

Since the relation R is reflexive, symmetric, and transitive, it is an equivalence relation